Multigraded commutative algebra

多项式方程是数学和科学中的一些最基本的对象，它们以大量的例子出现：费马最后定理、运动的物理模型、理想气体定律等等。因此，理解多项式方程的解是一个跨越所有量化领域的根本挑战。PI的工作旨在开发理论和算法方面的技术，以便更好地理解具有额外对称性的多项式方程的解。由于这类方程经常出现在数学或科学应用中，这项工作具有广泛影响的潜力。PI旨在扩展关于多阶多项式的文献，建立在他之前的工作基础上，使用虚拟分辨率作为多阶融合的几何应用的基础。一个项目将提供希尔伯特系统定理和贝林森对角线分解的新的多级类似物，从而建立两个基本结果。格林线性系统定理的多阶化版本的第二个项目将有更直接的几何应用。第三个项目将具有计算应用，产生一种新的、可能更快的算法来计算环簇上的束上同调。该项目还将通过PI对博士生的指导产生更广泛的影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Polynomial equations are some of the most fundamental objects in mathematics and science, and they arise in a huge array of examples: Fermat’s Last Theorem, physical models of motion, the ideal gas law, and much more. Understanding the solutions of polynomial equations is thus a fundamental challenge that cuts across all quantitative fields. The PI’s work aims to develop techniques—both theoretical and algorithmic—for better understanding the solutions of polynomial equations that are endowed with extra symmetries. As such equations often arise in mathematical or scientific applications, the work has the potential for wide impact.The PI aims to expand the literature on multigraded polynomials, building on his prior work of using virtual resolutions as a foundation for geometric applications of multigraded syzygies. One project would provide novel multigraded analogues of the Hilbert Syzygy Theorem and of Beilinson's resolution of the diagonal, thus establishing two foundational results. A second project on a multigraded version of Green's Linear Syzygy Theorem would have more direct geometric applications. And a third project would have computational applications, yielding a new and potentially much faster algorithm for computing sheaf cohomology on a toric variety. The project will also have broader impacts through the PI’s mentoring of PhD students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.